Higher order dynamic inequalities on time scales

Abstract

In this paper, we study dynamic inequalities where the domain of the unknown function is a so-called time scale T . The cases when the time scale equals to the reals or to the integers represent the classical theories of differential and of difference inequalities. A cover story article in New Scientist [24] discusses several other possible applications. Continuous and discrete inequalities are important in the analysis of qualitative properties of solutions of differential and difference equations [3, 18, 19] and as a result we believe that dynamic inequalities on time scales will be important in the analysis of qualitative properties of solutions of dynamic equations [20, 21]. In this paper, we will prove some new dynamic inequalities involving higher order on time scales. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [9]), i.e, when T = R, T = N and T = q0 = {qt : t ∈ N0} where q > 1. For more details of time scale analysis we refer the reader to the two books by Bohner and Peterson [6], [7] which summarize and organize much of the time scale calculus. The study of dynamic inequalities of Opial type on time scales was initiated by Bohner and Kaymakcalan [5] in 2001; see also the recent papers [10], [23] and [25] and the references cited therein. In [5] the authors showed that if x : [0,b]∩T → R is delta differentiable with x(0) = 0, then ∫ b